Are all embeddings of varieties essentially the same?

Question

Let $X$ be a nice projective variety, suppose we have two different embeddings to projective space $e_1, e_2$ of $X$ in $P^{n_1}, P^{n_2}$. Are there always embeddings $E_1: P^{n_1} \to P^{N}, E_2: P^{n_2} \to P^{N}$ so that $E_1\cdot e_1) = E_2\cdot e_2)$ ?

Answer 1

Let $L_i = e_i^*\mathcal{O}(1)$. These are very ample line bundles on $X$. If the embeddings $E_1$ and $E_2$ with the specified property exist then $L_1$ and $L_2$ are proportional in $\operatorname{Pic}(X)$. So, for a counterexample one needs a variety with non-cyclic Picard group.

One possible example is $X = \mathbb{P}^1 \times \mathbb{P}^1$ and the embeddings given by the very ample line bundles $$ L_1 = \mathcal{O}(1,1) \qquad\text{and}\qquad L_2 = \mathcal{O}(1,2). $$